Arbitrarily Accurate Analytical Approximations for the Error Function

TL;DR

This paper introduces spline-based integral approximations for the error function that converge faster than Taylor series, enabling highly accurate bounds, new series, and improved convergence properties for related functions.

Contribution

The paper presents novel spline-based approximation methods for the error function with arbitrarily high accuracy and applications to bounds, series, and nonlinear signal analysis.

Findings

Achieves a relative error bound of 1.43 x 10^-7 with four sub-intervals.

Provides approximations for error bounds at various accuracy levels.

Develops new series and functions related to the error function.

Abstract

In this paper a spline based integral approximation is utilized to propose a sequence of approximations to the error function that converge at a significantly faster manner than the default Taylor series. The approximations can be improved by utilizing the approximation erf(x) approximately equal to one for x>>1. Two generalizations are possible, the first is based on demarcating the integration interval into m equally spaced sub-intervals. The second, it based on utilizing a larger fixed sub-interval, with a known integral, and a smaller sub-interval whose integral is to be approximated. Both generalizations lead to significantly improved accuracy. Further, the initial approximations, and the approximations arising from the first generalization, can be utilized as the inputs to a custom dynamical system to establish approximations with better convergence properties. Indicative results include those of a fourth order approximation, based on four sub-intervals, which leads to a relative error bound of 1.43 x 10-7 over the positive real line. Various approximations, that achieve the set relative error bounds of 10-4, 10-6, 10-10 and 10-16, over the positive real, are specified. Applications include, first, the definition of functions that are upper and lower bounds, of arbitrary accuracy, for the error function. Second, new series for the error function. Third, new sequences of approximations for exp(-x2) which have significantly higher convergence properties that a Taylor series approximation. Fourth, the definition of a complementary demarcation function eC(x) which satisfies the constraint eC(x)^2 + erf(x)^2 = 1. Fifth, arbitrarily accurate approximations for the power and harmonic distortion for a sinusoidal signal subject to a error function nonlinearity. Sixth, approximate expressions for the linear filtering of a step signal that is modelled by the error function.